Uploaded by khanacademy on 27.10.2007

Transcript:

Welcome back

I am now going to do a series of videos on the trigonometric identities

So let's just review what we already know about the trig functions

so if I just write soh-cah-toa

soh-cah-toa

that tells us... and now that we've actually extended this with the unit circle definition... but if you watch those videos

you'll realize that the unit circle definition directly uses soh-cah-toa

so we'll just stick with soh-cah-toa

because I think it will help make some of what we are about to do seem a little bit more straightforward

and we'll kind of verge on the unit circle definition anyway

so we know that sine of theta

sine of theta is equal to Opposite over hypotenuse, right?

so, cosine of theta is equal to Adjacent over hypotenuse

and then the tangent of theta is equal to opposite over adjacent

so let's draw that out on a right triangle

we can do this with the unit circle as well

and it would make sense

let's see if we can find a relationship between sine, cosine and tangent

there's my right triangle

let's call this theta

this is the hypotenuse h

this is the opposite side, right? the opposite of theta

this is theta right here

this is the adjacent side, right?

well what do we know about the relationship between the opposite, the adjacent side and the hypotenuse?

what does the Pythagorean theorem tell us?

oh yeah, the opposite... this side squared plus this side squared is equal to the hypotenuse squared

so we can write that down

'a' squared plus 'o' squared is equal to the hypotenuse squared, right?

and then, this is just an equation, so if we want to we could divide both sides of this equation by h... by h squared

so what do we get?

we get 'a' squared over 'h' squared.. plus.. 'o' squared over 'h' squared is equal to one, right?

and then I could rewrite that as

('a' over 'h') squared plus ('o' over 'h') squared is equal to one

now do these look at all familiar?

hmm... what we have over here right?

this is 'a' over 'h'

and this is 'o' over 'h'

so we can just substitute

so this is just cosine of theta

cosine of theta squared and this is how you write cosine squared...

you could put parentheses round the whole thing and square it but this is just the notation people use

plus opposite over

so that's our first trig identity

so if you know the sine of theta it's very easy to figure out the cosine of theta, right?

you can just solve this equation

right if I know that.. I don't know, let's say I know that the sine of theta

let's say that I know the sine of theta is let's say one half, right?

then what is the cosine of theta?

cosine of theta is what?

well I know the sine of theta is one half, right so

cosine squared of theta plus

sine of theta is one half so

one half squared

is equal to one, right?

cosine squared theta plus one fourth is equal to one

so we have cosine squared theta is equal to three fourths

or cosine of theta would be the square root of this, we take the square root of both sides

the square root of three over two

and you probably remember that from our whole presentation on the 30-60-90 triangle

so I just wanted to show you a use of this trig identity

it's usually written sine squared plus cosine squared is equal to one

so let's just extend that one a little bit

let's play with the ratios and see what else we can... what other... identities... I guess, we can discover

oh whoops

clear "image invert colors"

so we know that sine squared theta plus cosine squared theta is equal to one

one thing we could do is divide both sides of this equation by cosine squared of theta

and let's just see what happens when we do that

so if we say... cosine squared theta... you have to distribute across... both terms

cosine squared theta... and then cosine squared of theta

well what's sine squared theta over cosine squared theta?

that's the same thing as ( sine of theta over cosine of theta ) squared

plus this is one

is equal to ( one over cosine theta ) squared

right, I just , I mean, one squared is one, I just rewrote it

so sine over cosine theta, I think we learned that already

that's just the tangent of theta

and in case you haven't learned that already, think about it this way

sine is opposite over the hypotenuse, right?

so that's opposite over hypotenuse

and then cosine is adjacent over hypotenuse

so adjacent over hypotenuse

and that equals opposite over hypotenuse times hypotenuse over adjacent

right, just dividing by a fraction is the same thing as multiplying by its reciprocal... that's all I did

and that equals opposite over adjacent, right?

so that equals opposite over adjacent

that just says sine of theta over cosine of theta is equal to tangent of theta

so sine squared theta over cosine squared theta is tan squared theta

and then plus one is ( one over cosine theta ) squared

and now I'm going to introduce a new trig ratio and it is really just one over cosine theta

one over cosine theta... I'm going to summarize this at the end just so that it's not too confusing

is actually the secant of theta

and it's just another ratio

the secant of theta instead of being the adjacent over the hypotenuse would be the hypotenuse over the adjacent, right?

it's just one over cosine theta, nothing famous, uh, nothing fancy here

so secant of theta

so that equals secant squared of theta

I know it can be a little overwhelming initially just because I'm throwing out all of these new terms

secant is one over cosine theta

but once you just play around with these enough and get familiar with the terms it'll make sense it'll be more natural to you

so this could be... you could view this as another trig identity

and actually in case... I don't even remember if I've taught it already you could view this as a trig identity... though that's almost definitional

and then of course you can, in case I haven't done it already

you now know that sine of theta over cosine of theta is equal to tangent of theta

and that's right here... I guess you could say the proof of it

so let me keep introducing you to more things

if this is really daunting maybe you can just rewatch it and hopefully it will make some sense

ah... let me see... clear.. image

so what have we learned so far?

we've learned that sine squared theta plus cosine squared theta is equal to one

we learned that sine of theta over cosine of theta is equal to tangent of theta

we learned that the tangent squared of theta plus one is equal to the secant of theta squared...

let me write this definition down...

oops... is equal to the secant squared of theta... sorry

and the secant of theta is just one over cosine of theta

and this is really something that you should just memorise

secant is one over cosine

and if you're wondering what one over sine is

one over sine of theta

it's the co-secant

the abbreviation is csc of theta

and if you're wondering what one over the tangent is

this is the co-tangent

and you just might want to memorize these

and this often confuses me... that one over the cosine is the secant but one over the sine is the cosecant

so it's almost the opposite, right?

one over the sine has a 'co' in it while one over the cosine doesn't have the 'co' in it

so, that might help you, uh, remember things

so I think that's all I have time for now

in the next presentation I'm going to introduce you to a couple more trig identities

see you soon